Topologically massive supersymmetric higher-spin actionsThese are higher-spin extensions of topologically massive \(\mathcal {N}=1\)  supergravity [1]}. involve different
massless sectors depending on the value of superspin. For integer superspin, the relevant massless actionThere is another massless action of this type, where the superfield \(H_{\alpha (2s)}\)  appears in the action with a first-order kinetic operator. However, the gauge transformations are of type B with \(t=1\) , \(\delta _{\Lambda }H_{\alpha (2s)}=\mathcal {D}_{\alpha (2)}\Lambda _{\alpha (2s-2)}\) . The corresponding action has been derived recently in [2]}. is first-order (in vector derivatives) and takes the form [3]}
\({\mathbb {S}}^{(2s)}_{\rm FO}[H,Y] &=&\bigg (-\frac{1}{2}\bigg )^s\frac{{\rm i}}{2}\int {\rm d}^{3|2} z \, E\,\Big \lbrace H^{\beta \alpha (2s-1)} {\cal D}^\gamma {\cal D}_\beta H_{\gamma \alpha (2s-1)} \nonumber \\&&+2{\rm i}(2s-1) Y^{\alpha (2s-2)} {\cal D}^{\beta (2)} H_{ \beta (2) \alpha (2s-2) } \nonumber \\&&+ (2s-1) \Big ( Y^{\alpha (2s-2) }{\cal D}^2 Y_{\alpha (2s-2)}+(2s-2){\cal D}_\beta Y^{\beta \alpha (2s-3) }{\cal D}^\gamma Y_{\gamma \alpha (2s-3)} \Big ) \nonumber \\&& -4{\cal S}{\rm i}\Big ( H^{\alpha (2s) } H_{\alpha (2s) }+2s(2s-1) Y^{\alpha (2s-2)} Y_{\alpha (2s-2)} \Big )\Big \rbrace  ~.\) 
